j.ph.piel (phd) application lab manager sopralab

EPFL October 7th 2008 1

Introduction to ellipsometry

J.Ph.PIEL (PhD)Application Lab ManagerSOPRALAB


OUTLINE :

- Basic Theory

- GES5 description

- Data Analysis

- Conclusion


I - BASIC THEORY

- Brief history of Ellipsometry

- Principle

- Physical meaning of ψ and Δ


Phase measurement

For the 1st time Paul DRUDE use ellipsometry in 1888

2000


Alexandre Rothen Published in 1945a paper where the word « Ellipsometer » appears for the 1st time

Thickness sensitivity : 0.3 A


Bref Historique

Schematic representation of the mounting from Alexandre Rothen (Rev. of scientifique instruments Feb 1945)


• Malus in 1808 : discovery of the polarisation of light by reflexion.

• Fresnel in 1823 : Wave theory of the light.

• Maxwell in 1873 : developpment of the Electrogmagnetic field theory.

• Drude in 1888 : Use of the extreme sensitivity of the ellipsometry to detect ultra thin layers (monolayer).

• Abeles in 1947 : Developpment of matrix formalism applied to thin layers stack.

• Rothe in 1945 : Introduction of the word : « Ellipsometer »

• Hauge, Azzam, Bashara in 1970 : description and developpement of different ellipsometer settings.

• 1980 : development of « Personal Computer » , automatisation of the technique. Industrial development of tools.

Few key people in ellipsometry area


INTRODUCTION

ELLIPSOMETRY is a method based on measurement of the change of thepolarisation state of light after reflection at non normal incidence on thesurface to study

-The measurement gives two independent angles: ψ and Δ

- It is an absolute measurement: do not need any reference- It is a non-direct technique: does not give directly the physicalparameters of the sample (thickness and index)- It is necessary to always use a model to describe the sample

SPECTROSCOPIC ELLIPSOMETRY (SE) gives more comprehensive results since it studies material on a wide spectral range


PRINCIPLE OF SPECTROSCOPIC ELLIPSOMETRY

ρ =rp

rs

= tan Ψ.e jΔ = f( ni, ki, Ti )

Substrate (ns, ks)

Thin Film 1 (n1, k1, T1)Thin Film 2 (n2, k2, T2)Thin Film i (ni, ki, Ti)

Ambient (n0, k0)

ES

EiEP

rs

rp

Erlinear polarisation

EiEpEs

ErEp.rpEs. rs

elliptical polarisation

θ0

Ellipsometry measures the complex reflectance ratio :


- After reflection on the sample, the extremity of the electric field vector describes an ellipse

εθ

0

rp

rs

p

s

- This ellipse is characterised by

*the ellipticity Tan ε which is the ratio of the large axis to the small axis

*the angle of rotation θ between the main axis and the P axis:

Ψ


Extremity of the electric field vector descrives an ellipse


Physical meaning of ψ and Δ.•Those parameters give all relevant information about the polarization state of the light at a given wavelength.

- Tan ψ gives exactly the angle of the first diagonal of the rectangle in which the ellipse is enclosed.

- Cos Δ gives roughly how fat is going to be the ellipse (shape).

Its mathematically linked to the ratio of short axis to long axis of the ellipse in its fundamental frame.

However, ψ and Δ are not straightforwardly linked to the easiest geometrical features ofthe ellipse.

θ : angle of rotation of the long axis of the ellipse versus axis p

Δ= CosTanTan )2()2( ψθ- 1 - 0.5 0.5 1

- 1

- 0.5

0.5

1

40.


- 1 - 0.5 0.5 1

- 1

- 0.5

0.5

1

0

- 1 - 0.5 0.5 1

- 1

- 0.5

0.5

1

20.

- 1 - 0.5 0.5 1

- 1

- 0.5

0.5

1

40.

- 1 - 0.5 0.5 1

- 1

- 0.5

0.5

1

60.

- 1 - 0.5 0.5 1

- 1

- 0.5

0.5

1

80.

- 1 - 0.5 0.5 1

- 1

- 0.5

0.5

1

90.

• Example of some different phase shift (Δ) for a given ψ value.• On those graphics, the long axis is the ellipse is represented and it has to be compared with the

diagonal of the rectangle.• Because ψ is fixed, the ellipse is enclosed in the same rectangle for each graphic.• Modification of Δ change both angle of inclination and ellipticity.


II - GES5 DESCRIPTION

- Physical description

- Jones Formalism

- Mathematical treatment of the signal

- Example


Goniometer

Analyser Arm

Polariser Arm

Sample

Mapping rho/theta

Microspots


SPECTROSCOPIC ELLIPSOMETER

Scanninigchannel

C.C.D.channel

Xe LampPA

Optical Fiber

Goniometer

Photo Multiplier

Tube


Spectrometer

Spectrograph

Schematics of dispersion elements

Grating : rotating

Prism : rotatingNIR detector

PMT

Entrance FiberGrating : fixed

MultichannelDetector

Entrance Fiber


GENERAL GES5 DESCRIPTION:

- Xenon Lamp:75 W, Short arc, High brilliance- MgF2 Rotating Polarizer MgF2 : 6 Hz - Adjustable Analyzer- Goniometer from 7° up to 90°- Microspots : spot size : 400 µm

- High resolution way :-Spectral range : 190 nm up to 2000 nm- resolution < 0.5 nm- double monochromator: prism + grating- Photon counting PMT

-High speed way : - Spectral range : 190 nm up to 1700 nm- Measurement duration : few seconds- Fixed grating spectrograph- CCD (1024x64 pixels)and OMA NIR (256 pixels) detectors


Jones Formalism

BUT : Jones formalism can only workif there is no depolarisation effects induced by the material


Optical system with no depolarisation effects is characterized by this following Jones Matrix :

2X2 complex Matrix


If the material is isotrope :

Ellipsometric Angles Ψ and Δ measured simultaneously


Pθ =R(-θ).Px.R(θ)

Jones Matrix for a Linear Polarizer : Px = ⎟⎟⎠

⎞⎜⎜⎝

⎛0001

Pθ = ⎟⎟⎠

⎞⎜⎜⎝

⎛− θθ

θθcossinsincos

Rotation Matix :

General expression for a polarizer where the main axis is oriented with an angle θ


I = Edp . Edp* + Eds . Eds

*

I (t) = I0 . ( 1 + α Cos 2 ω(t) + β Sin 2 ω(t) )A : Angle between Analyser and plane of Incidence.ω(t) : Angle between Polariseur and plane of Incidence.


Edp 1 0 Cos A Sin A rp 0 Cos P - Sin P 1 0 Ep

Eds 0 0 -Sin A Cos A 0 rs Sin P Cos P 0 0 Es

=

Detector Analyser Rotation Sample Rotation Polariser Lamp

A : Angle between Analyser axis and Plane of Incidence.P(t) : Angle between Polarizer axis and Plane of Incidence.

I = Edp . Edp* + Eds . Eds

*

I = I0 . ( 1 + α Cos 2 P(t) + β Sin 2 P(t) )


S1 S4S2 S3 S1

Time

Intensity

S I P dP10

4

= ∫ ( )π

S I P dP24

2

= ∫ ( )π

π

S I P dP32

34

= ∫ ( )π

π

S I P dP43

4

= ∫ ( )π

π

[S1 - S2 -S3 + S4 ]2 I0

α =[S1 + S2 - S3 - S4 ]

2 I0β =

[S1 + S2 + S3 + S4 ]Π

Ι0 =

HADAMART TRANSFORM


CALCULATION OF THE ELLIPSOMETRIC PARAMETERS

Cos 2 ATan 2 Ψ + Tan 2 A

Ι0 =

Tan 2 Ψ - Tan 2 ATan 2 Ψ + Tan 2 A

α = 2 Cos Δ . Tan Ψ. Tan ATan 2 Ψ + Tan 2 A

β =

1 + α 1 - α

Tan Ψ = Tan A .

β1 - α2

Cos Δ =


DUV UV VISIBLE NIR

ELLIPSOMETRIC MEASUREMENT


RESULTS ANALYSIS

Ti , ni , ki

Experimental Measurement=

Model Simulation ?

No

Yes

CosΔ

TanΨ

ExperimentalMeasurement

Physical ModelEstimated sample structure

- Film Stack and structure- Material n, k, dispersion- Composition Fraction of Mixture

REAL SAMPLE STRUCTURE

Non direct technique.

Need the use of modelsto interpret the measurements and to get physical parameters of the layers


III - DATA ANALYSIS

- Which physical parameters can we get ?

- Sensisitivy of the technique

- Description of the main models used

- How to describe optical properties of the materials.


- Mechanical thickness of each layer :Monolayer sensitivity (<0.01 A) up to 50 µm.

- Refractive index versus wavelength : n(λ)Allow to characterize the quality of the layer and the process.

- Extinction coefficient versus wavelength : k(λ).Measure the attenuation of light in the material.Allow to characterize the quality of the layer and the process.

- Surface and Interface characterization.

- Layer inhomogeneity characterization.

- In case of metallic or activated doping material:resistivity and doping concentration characterization using Drude Model.

- In case of porous material: porosity, pore size distribution and Young modulus characterization using Kelvin model.

Which physical parameters can we get ?


Spectroscopic ellipsometry sensitiviy

Phase variation : Δ is extremelly sensitive to ultra thin layers

Angle of incidence: 75°


0nm

10 A

Spectroscopic ellipsometry sensitiviy

Angle of incidence: 75°

Sensitivity could reach 0.01 Aor 1 picometer


Description of the main models used


2 media model : Substrat alone

Φ1

Φ0

Ambiant : air

Substratns = ñs - i.ks

rp = (ns.cosφ0 – n0.cosφs)/(ns.cosφ0 + ns.cosφ1)

and

rs= (n0.cosφ0 - ns.cosφs)/(ns.cosφ0 + ns.cosφ1)

withρ = rp/ rs = TanΨ.exp(i.Δ)

Fresnel equation and Snell-Descartes law:

ns = sinφ0.(1 + ((1-ρ)/(1+ρ))2.Tan2φ0)1/2

Direct inversion of the ellipsometric parameters to get substrate indices

EPFL October 7th 2008 37Silicium Angle of incidence 75°

2 media model : Substrat alone


3 media model : one layer on known substrate


Native oxide (SiO2) on SiliconSicr

SiO2 30.8 Å



Sicr

SiO2 1200 Å

SiO2 layer on silicon




Thick SiO2 layer on Silicon Sicr

SiO2 1,9838µm


Rp et Rs are periodic function Same periods

Idem for Tan Ψ and Cos Δ

If φa =75°Period=λ/(nb

2-0.93)0.5

For SiO2 filmnb =1.5 Period = λ/2.3

At λ=450 nm, period = 200 nm

Sensitivity to Δ :

In this caseΔ variation de 360° for a period = 200 nm

Δ sensitivity given by instrument: 5.10-2°

Thickness sensitivity : 0.01 A



Multilayer stack


Interface relations : FresnelPropagation inside the layer :Interferences

-

Multilayer stack


How to describe optical properties of the materials

• Index library• Effective medium mixing laws• Dispersion law• Harmonic oscillators laws• Drude laws


Index Library3 main type of materials:

Dielectric : transparent in the visible range but absorbing in the UV and have absorbing band in the IR.Transparent materials : Oxides (SiO2, TiO2) Fluorides (MgF2)Optical filters, Anti reflective coatings, dielectric mirror (lasers).

Semi-conductors dispersives laws extremelly rich in the visible range linked to the band structures. Could be metallic in the IR.Silicon : Si; Germanium : Ge; Gallium arsenide : GaAs;Gallium nitride: GaN;Carbon Silicon SiC (blue diode);

Metal highly absorbing in the visible. Infrared mirrors :Au, Al, CuMagnetic Materials : Co, Ni, Fe, Gd

• Handbook of Optical Constants from Palick• SOPRA library• Direct measurements on bulk materials


Effective medium model


Popular effective medium model


Interface and Surface roughnesstreated by Effective medium Model


- Model is convenient for physical or mechanical mixing.Ex : Porous material with inclusion of void.

- Model is not convenient for chemical mixingEx : Inclusion of atom in elementary cellsor variable atomic concentration Ex : Si(1-x)Gex.

- Model is not applicable when the size of inhomogeneitiesexceed few hundredth (1/100) of the wavelenght of the beam.

Limits of the effective medium model


Dispersion LawSellmeier law :

42 λλCBAn ++= 53 λλλ

FEDk ++=

Cauchy law :

k =0 for transparent materials

)()1(1)(

2

22

BA

r −∗−

+=λ

λλε53

)(λλλ

λε EDCi ++=

1

3

321

4

2

2

3

2

2

2

10 ).(.)()(..)(

−

⎥⎦⎤

⎢⎣⎡ ++=

−+

−+=

λλλλλ

λλ

λλλ BBBnk

AA

AAAn

SiNx example


4 absorption peaks

SiO2 in the Infrared range2222

0

2

3

2222

0

2

2

0

22

.)(..

.)()(..

λλλε

λλλλε

Γ+−Γ

=Γ+−−

=LA

LLA

ir

Harmonic Oscillators

Absorption band in the measured spectral rangeLorentz oscillators

A: Intensity.

L0: Middle wavelength.

Γ : Oscillator width.


100 nm of an absorbing layer on Silicon substrate

Spectroscopic Ellipsometry data Indices of the layer

Harmonic Oscillators


Wavelength ( μm)

2 4 6 8 10 12 14 16

n, k

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

Measure and Drude lawfit on doped silicon

undoped silicon

( )ε ω εω

ω ωτ1

2

2 2= −−∞

p

( )ε ωω ω

ω ω ωτ

τ2

2

2 2=−

p

( )

Indices are fitted using Drude law :

ωp : Plama frequencyωτ : diffusion frequency

Semi conductors indices are sensitive to the doping level in the IR range

Drude Law

N(λ) doped SiliconN(λ)

k(λ) doped Siliconk(λ) undoped Silicon


μωτ

=e

m*

Spectroscopic ellipsometry fit gives :

• Plasma frequency ωp and Diffusion Frequency ωτ

• Material Conductivity

• Carrier Density

• Carrier Mobility μ

σ εωω τ

= 0

2p

N me

p= * ε ω02

2

For the sample corresponding to the previous measurement :

• N =1.6 1019 at./cm3

• μ = 104 cm2 V-1 s-1

• σ = 264 Ω-1 cm-1

Drude Law


Conclusion


The measurement gives two independent angles: ψ and Δ

- Absolute measurement: do not need any reference.- Extremely sensitive to very thin layers (less than a monolayer).- fast : get the full spectrum (190 nm up 1700 nm) in few seconds

- Non-direct technique: does not give directly the physicalparameters of the sample (thickness and index)- Need to use a model to describe the sample.

Determination of : but also :Thickness PorosityRefractive index : n ResistivityExtinction coefficient : k Molecular bounds

j.ph.piel (phd) application lab manager sopralab

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